Improving a primal–dual simplex-type algorithm using interior point methods

Abstract

Interior point methods and simplex-type algorithms are the most widelyused algorithms for solving linear programming problems. The simplex algorithm has many important applications. Hence, even small improvements in simplex-type algorithms could result in noticeable practical impact. This paper presents a hybrid algorithm that combines the strengths of interior point methods and exterior point simplex algorithms. It applies an interior point method for a few iterations leading to significant improvement of the objective function value. At this point, the proposed algorithm uses an exterior point simplex algorithm to find an optimal solution. A crucial point is the selection of the interior point that will be used by the exterior point simplex algorithm to calculate a direction to the feasible region. The goal of the proposed implementation is twofold: (i) improve the performance of the exterior point algorithm, and (ii) find an optimal basic solution starting from an interior point (purification process). The latter goal is very important since an optimal basic solution can be used to solve closely related linear programming problems (warm-start) and linear programming relaxations of integer programming problems. Computational results on a set of benchmark problems (Netlib, Kennington, Mészáros) are presented to demonstrate the efficiency of the proposed hybrid algorithm. The results show that the proposed algorithm is 1.53× faster than the exterior point simplex algorithm.